# 7.1 Digital receiver: carrier recovery  (Page 2/3)

 Page 2 / 3

## Phase detector

The goal of the PLL is to maintain a demodulating sine and cosine that match the incoming carrier. Suppose ${\omega }_{c}$ is the believed digital carrier frequency. We can then represent the actual received carrier frequency as theexpected carrier frequency with some offset, $\stackrel{˜}{{\omega }_{c}}={\omega }_{c}+\stackrel{˜}{\theta }(n)$ . The NCO generates the demodulating sine and cosine with the expected digital frequency ${\omega }_{c}$ and offsets this frequency with the output of the loop filter. The NCO frequency can then be modeled as $\stackrel{^}{{\omega }_{c}}={\omega }_{c}+\stackrel{^}{\theta }(n)$ . Using the appropriate trigonometric identities $\cos A\cos B=1/2(\cos (A-B)+\cos (A+B))$ and $\cos A\sin B=1/2(\sin (B-A)+\sin (A+B))$ . , the in-phase and quadrature signals can be expressed as

${z}_{0}(n)=1/2(\cos (\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n))+\cos (2{\omega }_{c}+\stackrel{˜}{\theta }(n)+\stackrel{^}{\theta }(n)))$
${z}_{Q}(n)=1/2(\sin (\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n))+\sin (2{\omega }_{c}+\stackrel{˜}{\theta }(n)+\stackrel{^}{\theta }(n)))$
After applying a low-pass filter to remove the double frequency terms, we have
${y}_{1}(n)=1/2\cos (\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n))$
${y}_{Q}(n)=1/2\sin (\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n))$
Note that the quadrature signal, ${z}_{Q}(n)$ , is zero when the received carrier and internallygenerated waves are exactly matched in frequency and phase. When the phases are only slightly mismatched we can use therelation
$\forall \theta , \mathrm{small}\colon \sin \theta \approx \theta$
and let the current value of the quadrature channel approximate the phase difference: ${z}_{Q}(n)\approx \stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n)$ . With the exception of the sign error, this difference is essentially how much we need to offset our NCOfrequency If $\stackrel{˜}{\theta }(n)-\stackrel{^}{\theta }(n)> 0$ then $\stackrel{^}{\theta }(n)$ is too large and we want to decrease our NCO phase. . To make sure that the sign of the phase estimate is right, in this example the phase detector issimply negative one times the value of the quadrature signal. In a more advanced receiver, information from boththe in-phase and quadrature branches is used to generate an estimate of the phase error. What should the relationship between the I and Q branches be fora digital QPSK signal?

## Loop filter

The estimated phase mismatch estimate is fed to the NCO via a loop filter, often a simple low-pass filter. For thisexercise you can use a one-tap IIR filter,

$y(n)=\beta x(n)+\alpha y(n-1)$
To ensure unity gain at DC, we select $\beta =1-\alpha$

It is suggested that you start by choosing $\alpha =0.6$ and $K=0.15$ for the loop gain. Once you have a working system, investigate the effects of modifying these values.

## Matlab simulation

Simulate the PLL system shown in [link] using MATLAB. As with the DLL simulation, you will have to simulate the PLL on a sample-by-sample basis.

Use [link] to implement your NCO in MATLAB. However, to ensure that the phase term does not grow toinfinity, you should use addition modulo $2\pi$ in the phase update relation. This can be done by setting $\theta (n)=\theta (n)-2\pi$ whenever $\theta (n)> 2\pi$ .

[link] illustrates how the proposed PLL will behave when given a modulated BPSK waveform. In this case thetransmitted carrier frequency was set to $\stackrel{˜}{{\omega }_{c}}=\frac{\pi }{2}+\frac{\pi }{1024}$ to simulate a frequency offset.

Note that an amplitude transition in the BPSK waveform is equivalent to a phase shift of the carrier by $\frac{\pi }{2}$ . Immediately after this phase change occurs, the PLL begins to adjust the phase to force the quadraturecomponent to zero (and the in-phase component to $1/2$ ). Why would this phase detector not work in a real BPSK environment? How could it be changed to work?

what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!